Chapter 4, Problem 82P

(a)

To determine

The acceleration of each of the blocks.

Explanation of Solution

Introduction:

Newton’s third law states that for every action, there is an equal and opposite reaction. It means for every interaction between two objects, there is a pair of forces that act on both the objects.

The free-body diagram for both the block is shown in Figure 1

  

Figure 1

In the above diagram is shown that the force on the block $m_1$ is the gravitational force and the force due to tension on the string. In the same way, the force on the block of mass $m_2$ is the gravitational force and the force due to tension on the string.

Conclusion:

Therefore, the free body diagram for both blocks is shown in Figure 1

(b)

To determine

The expression for the acceleration of the block and the tension in the string.

Answer to Problem 82P

The expression for the acceleration is $\frac{\left(m_1-m_2\right)g}{m_1+m_2}$ and the expression for the tension in the string is $\frac{2m_1{m}_2}{m_1+m_2}g$ . $$

Explanation of Solution

Calculation:

The equation of motion for the block $m_1$ is given by,

  $\begin{array}{l}{m}_{1}a=m_{1}g-T\\ T=m_{1}g-m_{1}a\end{array}$

The equation of motion for the block $m_2$ is given by,

  $\begin{array}{l}{m}_{2}a=T-m_{2}g\\ T=m_{2}a+m_{2}g\end{array}$

The expression for the acceleration of the block is evaluated as,

  $\begin{array}{l}{m}_{1}g-m_{1}a=m_{2}a-m_{2}g\\ m_{1}g-m_{2}g=m_{1}a+m_{2}a\\ a=\frac{\left(m_1-m_2\right)g}{m_1+m_2}\end{array}$

The expression for the tension in the string is evaluated a,

  $\begin{array}{c}T=m_{1}g-m_{1}a\\ =m_{1}g-m_1\frac{\left(m_1-m_2\right)g}{m_1+m_2}\\ =m_{1}g\left[\frac{m_1+m_2-m_1+m_2}{m_1+m_2}\right]\\ =\frac{2m_1{m}_2}{m_1+m_2}g\end{array}$

Conclusion:

Therefore, the expression for the acceleration is $\frac{\left(m_1-m_2\right)g}{m_1+m_2}$ and the expression for the tension in the string is $\frac{2m_1{m}_2}{m_1+m_2}g$ . $$

(c)

To determine

Whether the expressions for the tension and the acceleration gives the plausible results if $m_1=m_2$ in the limit that $m_1\gg m_2$ and in the limit that $m_1\ll m_2$ .

Answer to Problem 82P

The value of tension for the condition $m_1=m_2$ is $m_{1}g$ and the value of acceleration is $0$ . For the value of $m_1\gg m_2$ the plausible value of the tension is $2m_{2}g$ and the value of acceleration is $g$ . For the value of $m_1\ll m_2$ , the plausible value of the tension is $2m_{1}g$ and the value of acceleration is $-g$ .

Explanation of Solution

Calculation:

For the value of $m_1=m_2$ the value of $m_2$ is negligible and the expression for the acceleration can be evaluated as,

  $\begin{array}{c}a=\frac{\left(m_1-m_2\right)}{m_1+m_2}g\\ =0\end{array}$

For the value of $m_1=m_2$ the value of $m_2$ is negligible and the expression for the tension can be evaluated as,

  $\begin{array}{c}T=\frac{2m_1{m}_2}{m_1+m_2}g\\ =m_{1}g\end{array}$

For the value of $m_1\gg m_2$ the value of $m_2$ is negligible and the expression for the acceleration can be evaluated as,

  $\begin{array}{c}a=\frac{\left(m_1-m_2\right)}{m_1+m_2}g\\ =\frac{m_1}{m_2}g\\ =g\end{array}$

For the value of $m_1\gg m_2$ the value of $m_2$ is negligible and the expression for the tension can be evaluated as,

  $\begin{array}{c}T=\frac{2m_1{m}_2}{m_1+m_2}g\\ =2m_{2}g\end{array}$

For the value of $m_1\ll m_2$ the value of $m_1$ is negligible and the expression for the acceleration can be evaluated as,

  $\begin{array}{c}a=\frac{\left(m_1-m_2\right)}{m_1+m_2}g\\ =-\frac{m_2}{m_2}g\\ =-g\end{array}$

For the value of $m_1\ll m_2$ the value of $m_1$ is negligible and the expression for the acceleration can be evaluated as,

  $\begin{array}{c}T=\frac{2m_1{m}_2}{m_1+m_2}g\\ =2m_{1}g\end{array}$

Conclusion:

Therefore, the value of tension for the condition $m_1=m_2$ is $m_{1}g$ and the value of acceleration is $0$ . For the value of $m_1\gg m_2$ the plausible value of the tension is $2m_{2}g$ and the value of acceleration is $g$ . For the value of $m_1\ll m_2$ , the plausible value of the tension is $2m_{1}g$ and the value of acceleration is $-g$ .